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Chapter 7: Problem 56

Prove that the Trapezoid Rule is exact (no error) when approximating the definite integral of a linear function.

Short Answer

Expert verified
Explain. Answer: Yes, the Trapezoid Rule is exact when used to approximate the definite integral of a linear function. This is because, when applied to a linear function, the Trapezoid Rule's approximation is equal to the exact definite integral value, resulting in no error.

Step by step solution

01

Understanding the Trapezoid Rule

The Trapezoid Rule is a numerical integration method designed to approximate the definite integral of a function, f(x), over a given interval [a, b]. The idea behind the Trapezoid Rule is to approximate the area under the curve as a trapezoid, with a height given by the average value of the function at the endpoints of the interval. The formula for the Trapezoid Rule is: \[ \int_a^b f(x) dx \approx \frac{b-a}{2} \left[f(a)+f(b)\right] \] Given a linear function, say f(x) = mx+c, we will use the Trapezoid Rule to approximate its definite integral over an interval [a, b] and show that there's no error in the approximation.
02

Find the definite integral of the linear function

We will start by finding the definite integral of the given linear function, f(x) = mx+c, over the interval [a, b]. Calculating the definite integral, we get: \[ \int_a^b (mx+c) dx = \left[\frac{1}{2}mx^2+cx \right]_a^b \] Now, we will evaluate this expression at the endpoints of the interval (a and b) and subtract the values to get the exact definite integral value. \[ \left(\frac{1}{2}mb^2+cb\right) - \left(\frac{1}{2}ma^2+ca\right) = \left(\frac{m}{2}(b^2-a^2) + c(b-a)\right) \] This expression represents the exact value of the linear function's definite integral on the interval [a, b].
03

Apply the Trapezoid Rule to the Linear Function

Now, we will apply the Trapezoid Rule to the linear function on the interval [a, b]. Using the formula mentioned in step 1: \[ \int_a^b f(x) dx \approx \frac{b-a}{2} \left[f(a)+f(b)\right]= \frac{b-a}{2} [(ma+c)+(mb+c)] \] Simplify the expression: \[ \frac{b-a}{2} (ma+c+mb+c) = \left(\frac{m}{2}(b^2-a^2) + c(b-a)\right) \]
04

Compare the Exact Integral and the Trapezoid Rule Approximation

In step 2, we found the exact definite integral value to be: \[ \left(\frac{m}{2}(b^2-a^2) + c(b-a)\right) \] In step 3, we found the Trapezoid Rule approximation to be: \[ \left(\frac{m}{2}(b^2-a^2) + c(b-a)\right) \] Since both the exact definite integral value and the Trapezoid Rule approximation are equal, we can conclude that the Trapezoid Rule is exact (no error) when approximating the definite integral of a linear function.

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Most popular questions from this chapter

Use the following three identities to evaluate the given integrals. $$\begin{aligned}&\sin m x \sin n x=\frac{1}{2}[\cos ((m-n) x)-\cos ((m+n) x)]\\\&\sin m x \cos n x=\frac{1}{2}[\sin ((m-n) x)+\sin ((m+n) x)]\\\&\cos m x \cos n x=\frac{1}{2}[\cos ((m-n) x)+\cos ((m+n) x)]\end{aligned}$$ $$\int \sin 5 x \sin 7 x d x$$

The following integrals require a preliminary step such as long division or a change of variables before using partial fractions. Evaluate these integrals. $$\int \frac{d x}{1+e^{x}}$$

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Find the volume of the described solid of revolution or state that it does not exist. The region bounded by \(f(x)=(x+1)^{-3 / 2}\) and the \(y\) -axis on the interval (-1,1] is revolved about the line \(x=-1.\)

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